Multi-scale computational framework for simulation of cellular solids

Xiaowo Wang, Purdue University

Abstract

Natural and man-made cellular solids have been used in a variety of engineering applications. Despite their widespread use, the behavior of such materials is not well understood because of their high heterogeneity and complex micro-structure. Conventionally, these materials have been studied with experimental testing to obtain rather simplistic models and empirical relationships describing their behavior. However, most existing methods are limited to the small deformation regime and almost none of the mechanics-based models are able to capture the behavior of cellular solids under moderate-to-high strain-rate loadings commonly encountered in practical applications such as impact absorption and blast mitigation. In this research, a multi-scale computational framework is formulated to study the behavior of structures and components made from cellular solids over a wide range of spatial and temporal scales. The framework consists of two levels of models, a continuum level and a micro-structural level. The continuum-level models utilize finite element and mesh-free methods in conjunction with spatial domain decomposition and temporal multi-time-stepping to capture different local- and global-scale behavior. At the micro-structural level, the framework relies on a realistic representation of the foam micro-structure and homogenization techniques to couple it with the continuum-level models. This research focuses on addressing different aspects of the formulation and implementation of this multi-scale framework. At the continuum-level, a variationally consistent coupling method is formulated for coupling subdomains that use different non-matching discretizations. It is shown that, while existing coupling methods in the literature are unable to pass patch tests in general, the variationally consistent method is able to pass arbitrary orders of patch tests exactly to within numerical precision for any numerical integration scheme that may be used for the subdomains. The method also shows good convergence for problems in general. Several numerical problems are solved using this method to compare its performance against existing coupling methods. For modeling of temporal multi-scale phenomena, two multi-time-step coupling methods are formulated. These methods allow one to use different time-steps to simulate different subdomains within a large structural model. A modified multi-time-step method for Newmark time integration schemes is formulated and shown to improve the computational efficiency of an existing consistent multi-time-step method. Another multi-time-step method is formulated for the Bathe time integration scheme and shown to work well for highly dynamic problems such as impact and wave propagation. At the micro-structural scale, a reduced-order model consisting of rotational and translational spring elements that simulate the behavior of individual ligaments of an open-cell foams is utilized. Using a computational homogenization approach, the micro-structural forces are up-scaled to the continuum-level to obtain macro-stress. A secant method is used to approximate the tangent from the micro-structural model for iterating the continuum-level solution to convergence. The performance of this micro-structural model is verified using the problem of crushing of a sample of open-cell foam.

Degree

Ph.D.

Advisors

Prakash, Purdue University.

Subject Area

Civil engineering|Materials science

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